Where we disagree with ourselves
More liquid: our numbers and our cooks point opposite ways
This is the honest state of the evidence and we would rather
print it than pick the half we like.
Adding extra liquid (water or broth) beyond the recipe amount usually thinned the sauce as expected (values of 1, 3, 3, 3, 4, 4 out of 10 across six reports), but in three other reports the sauce stayed thick despite the added liquid (7-9/10) -- total reduction time and how much was added evidently matter more than the simple presence of extra liquid.
6 cooks report this, 4 disagree
Against that, our own fitted model of 25 within-recipe comparisons finds
more braising liquid going with a higher sauce-body score
(q = 0.0004). A within-recipe comparison is the harder one to
explain away: it is the same recipe, cooked by different people, some of
whom added liquid. If it were simply that wetter recipes happen to be
better recipes, this fit should not survive.
Diluting the stock ratio with more water -- fewer bouillon cubes, half broth/half water -- without upping the thickener leaves the sauce noticeably thinner; a modest splash of extra liquid alone doesn't thin it as much.
3 cooks report this, 3 disagree
The most likely reconciliation, and it is a hypothesis rather than a
result: total reduction time matters more than the volume that went in, so
a cook who adds a cup and then simmers it away is not doing what a cook who
adds a cup at the end is doing. Our data cannot separate those two, because
nobody records how long they reduced. A controlled test would clear it up
in an afternoon, and it is the single most useful test anybody could run on
this dish.
Until then: correct thickness by reducing rather than by
adding, which is the direction every line of evidence here agrees is
safe.